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First of all, rest assured that I have used the search tool and read Why do we require a topological space to be closed under finite intersection? and https://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets.

I'm looking for a more logical approach on this issue. I understand intuitively the reasons why an intersection of infinitely many open sets may fail to be open, but only in the case of a topology induced by a metric. In the general sense, what would be a somewhat rigorous argument? I feel that some counterexamples aren't enough for me to deeply understand the motives. I've read somewhere that the intersection of an infinite family of sets is related with an infinite conjunction, which is not allowed in the usual logical framework we use for mathematics. Is this the issue? Could you explain it in more detail? Are infinite disjunctions permissible, then? Finally, if we were to work within an infinitary logic, would this "issue" disappear, and consequently, would the building of topology become somewhat trivial, or less rich and interesting?

Thanks.

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Abel
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    What do you mean by "rigorous argument"? We're talking about a definition (the definition of "topology"); examples and counterexamples are precisely what guide definitions. You cannot logically demonstrate a definition. – Arturo Magidin Mar 31 '11 at 20:50
  • Unions are related to disjunctions, not conjunctions; intersections are related to conjunctions. This because $x\in\cup_{i\in I} A_i$ if and only if $\lor_{i\in I} x\in A_i$ is true. – Arturo Magidin Mar 31 '11 at 20:53
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    I don't understand what you mean by "logical." A counterexample is a counterexample. Once you find one, you don't need to work more to find others to disprove what you're trying to disprove. – Qiaochu Yuan Mar 31 '11 at 20:57
  • @Arturo: I'm referring to the relation I think these concepts have with infinite conjunctions. I did say "somewhat" :) While I understand that in some sense the examples and counterexamples are the fundamental motivation for the definitions, it's the relation between the definition of a topology and the logical issue of long statements what I'm particularly interested about. Of course, I'm not entirely sure about any of this, but I believe I've read mentions of this connection. – Abel Mar 31 '11 at 21:00
  • @Arturo: About the conjunction/disjunction mix-up, I'll edit. – Abel Mar 31 '11 at 21:01
  • @Qiaochu: I understand that, but I don't find a counterexample very enlightening. For example, if I claimed that $x^2 = -2$ for all real $x$, and then found that when $x$ is $3$ I have a counterexample, I would know that my claim was false, but I wouldn't be seeing the deeper motives. In this question I'm wondering about the connection, if it exists, between the size of families of sets that unions and intersections can be taken over and infinitely long logical statements. – Abel Mar 31 '11 at 21:10
  • @Abel: in that case, I don't see how the question you're asking isn't answered by the MO threads. – Qiaochu Yuan Mar 31 '11 at 21:38
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    @Abel: What "logical issue of long statements"? An arbitrary union may be seen to correspond to an arbitrary disjunction, an arbitrary intersection to an arbitrary conjunction. But you do realize that one can equally well define a topology in terms of its closed sets, instead of its open sets? If you think "arbitrary union" is fine because it somehow "is" a disjunction and "arbitrary intersection" isn't because it somehow "is" a conjunction, then how would that square with defining topology using closed sets, where you would have "arbitrary intersection* but not "arbitrary union"? – Arturo Magidin Mar 31 '11 at 21:57

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If you defined arbitrary infinite intersections of open sets to be open, then in any Hausdorff space, every subset would be open --- thus making the definition completely useless. To see this, note that the intersection of all open sets containing a point $x$ would be the singleton set $\{x\}$ --- thus all singleton sets would be open. But, recalling that arbitrary unions of open sets is open, any set $X$ would be expressible as a union of singleton open sets $X=\bigcup\{\{x\}:x\in X\}$.

Apollo
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Regarding your points about logic: there is a logic called geometric logic wherein we have finite conjunctions, infinitary disjunctions, existential quantification, infinitary distributivity, and the Frobenius axiom. It is more or less the logic you get your truth-values are the open sets of a topological space.

Geometric logic not quite the same as even, say, intuitionistic first-order logic: there are theories which can be expressed in one but not the other, e.g. the theory of rings with nilpotent elements is geometric, but not first-order. I'm afraid this doesn't really answer your question though — these facts were discovered after the axioms of topology were laid down.

Zhen Lin
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I'm not sure what you mean by a logical reason, but maybe the following is helpful. Essentially I'm going to suggest that it makes sense if we want to think about limits; otherwise we kind of force any space to either have isolated points or infinitesimally close elements.

We want somehow to capture the idea of closeness without having to refer to distance. Imagine you want to be able to talk about a sequence of points $(p_0, p_1, ...)$ in $X$ approaching a point $x \in X$. The intuition is something like that, for any collection of points which is a collection of points "near to" $x$ that we pick, we expect our sequence to eventually enter this collection and stay in it. That is, we want to talk about some family of "neighborhoods" (nearby areas) of $x$, and we want to say that for any neighborhood of $x$, the sequence eventually stays in it.

Now, suppose we require that every arbitrary intersection of neighborhoods of $x$ still be considered a neighborhood of $x$. Then there are two possibilities. One possibility is that we can take some intersection and get $\{ x\}$ as a neighborhood. In this case, no sequence of other elements can converge to $x$. The second possibility is that there must be a distinct point $y\neq x$ such that $y\in U$ for every neighborhood $U$ of $x$ (because even the intersection over all neighborhoods of $x$ is not just ${ x}$). In this case, if we want to keep thinking of a neighborhood as a collection of nearby points, we're essentially viewing $y$ as infinitely close to $x$.

So, since we might sometimes want to avoid having infinitesimally close elements (i.e. indistinguishable by the topology) and isolated points, we shouldn't require that every arbitrary intersection still be considered a neighborhood.

matt
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